Kahan p9 Example

Percentage Accurate: 69.2% → 100.0%

Time: 12.1s

Alternatives: 12

Speedup: 2.5×

Specification

\[\left(0 < x \land x < 1\right) \land y < 1\]

Math

\[\begin{array}{l} \\ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))

C

double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x - y) * (x + y)) / ((x * x) + (y * y))
end function

Java

public static double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

Python

def code(x, y):
	return ((x - y) * (x + y)) / ((x * x) + (y * y))

Julia

function code(x, y)
	return Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
end

Wolfram

code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))

C

double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x - y) * (x + y)) / ((x * x) + (y * y))
end function

Java

public static double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

Python

def code(x, y):
	return ((x - y) * (x + y)) / ((x * x) + (y * y))

Julia

function code(x, y)
	return Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
end

Wolfram

code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{x - y\_m}{\mathsf{hypot}\left(x, y\_m\right) \cdot \frac{\mathsf{hypot}\left(x, y\_m\right)}{x + y\_m}} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (/ (- x y_m) (* (hypot x y_m) (/ (hypot x y_m) (+ x y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return (x - y_m) / (hypot(x, y_m) * (hypot(x, y_m) / (x + y_m)));
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return (x - y_m) / (Math.hypot(x, y_m) * (Math.hypot(x, y_m) / (x + y_m)));
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return (x - y_m) / (math.hypot(x, y_m) * (math.hypot(x, y_m) / (x + y_m)))

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(Float64(x - y_m) / Float64(hypot(x, y_m) * Float64(hypot(x, y_m) / Float64(x + y_m))))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = (x - y_m) / (hypot(x, y_m) * (hypot(x, y_m) / (x + y_m)));
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(x - y$95$m), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] * N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] / N[(x + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.5%

   \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 76.5%

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}} \]

   2. times-frac 76.6%

      \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}} \]

   3. hypot-define 76.6%

      \[\leadsto \frac{x - y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}} \]

   4. hypot-define 99.9%

      \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}} \]
5. Step-by-step derivation

   1. clear-num 99.9%

      \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}} \]

   2. frac-times 99.9%

      \[\leadsto \color{blue}{\frac{\left(x - y\right) \cdot 1}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}}} \]

   3. *-rgt-identity 99.9%

      \[\leadsto \frac{\color{blue}{x - y}}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}} \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}}} \]

7. Final simplification 99.9%

   \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}} \]
8. Add Preprocessing

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{x - y\_m}{\mathsf{hypot}\left(x, y\_m\right)} \cdot \frac{x + y\_m}{\mathsf{hypot}\left(x, y\_m\right)} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (* (/ (- x y_m) (hypot x y_m)) (/ (+ x y_m) (hypot x y_m))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return ((x - y_m) / hypot(x, y_m)) * ((x + y_m) / hypot(x, y_m));
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return ((x - y_m) / Math.hypot(x, y_m)) * ((x + y_m) / Math.hypot(x, y_m));
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return ((x - y_m) / math.hypot(x, y_m)) * ((x + y_m) / math.hypot(x, y_m))

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(Float64(Float64(x - y_m) / hypot(x, y_m)) * Float64(Float64(x + y_m) / hypot(x, y_m)))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = ((x - y_m) / hypot(x, y_m)) * ((x + y_m) / hypot(x, y_m));
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(x - y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x + y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.5%

   \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 76.5%

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}} \]

   2. times-frac 76.6%

      \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}} \]

   3. hypot-define 76.6%

      \[\leadsto \frac{x - y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}} \]

   4. hypot-define 99.9%

      \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}} \]

5. Final simplification 99.9%

   \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)} \]
6. Add Preprocessing

Alternative 3: 93.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := x \cdot x + y\_m \cdot y\_m\\ \mathbf{if}\;\frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{t\_0} \leq 2:\\ \;\;\;\;\frac{\left(x \cdot y\_m - y\_m \cdot y\_m\right) + x \cdot \left(x - y\_m\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y\_m}{\mathsf{hypot}\left(x, y\_m\right)} \cdot \left(1 + \frac{x}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (+ (* x x) (* y_m y_m))))
   (if (<= (/ (* (- x y_m) (+ x y_m)) t_0) 2.0)
     (/ (+ (- (* x y_m) (* y_m y_m)) (* x (- x y_m))) t_0)
     (* (/ (- x y_m) (hypot x y_m)) (+ 1.0 (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = (x * x) + (y_m * y_m);
	double tmp;
	if ((((x - y_m) * (x + y_m)) / t_0) <= 2.0) {
		tmp = (((x * y_m) - (y_m * y_m)) + (x * (x - y_m))) / t_0;
	} else {
		tmp = ((x - y_m) / hypot(x, y_m)) * (1.0 + (x / y_m));
	}
	return tmp;
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = (x * x) + (y_m * y_m);
	double tmp;
	if ((((x - y_m) * (x + y_m)) / t_0) <= 2.0) {
		tmp = (((x * y_m) - (y_m * y_m)) + (x * (x - y_m))) / t_0;
	} else {
		tmp = ((x - y_m) / Math.hypot(x, y_m)) * (1.0 + (x / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = (x * x) + (y_m * y_m)
	tmp = 0
	if (((x - y_m) * (x + y_m)) / t_0) <= 2.0:
		tmp = (((x * y_m) - (y_m * y_m)) + (x * (x - y_m))) / t_0
	else:
		tmp = ((x - y_m) / math.hypot(x, y_m)) * (1.0 + (x / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(x * x) + Float64(y_m * y_m))
	tmp = 0.0
	if (Float64(Float64(Float64(x - y_m) * Float64(x + y_m)) / t_0) <= 2.0)
		tmp = Float64(Float64(Float64(Float64(x * y_m) - Float64(y_m * y_m)) + Float64(x * Float64(x - y_m))) / t_0);
	else
		tmp = Float64(Float64(Float64(x - y_m) / hypot(x, y_m)) * Float64(1.0 + Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = (x * x) + (y_m * y_m);
	tmp = 0.0;
	if ((((x - y_m) * (x + y_m)) / t_0) <= 2.0)
		tmp = (((x * y_m) - (y_m * y_m)) + (x * (x - y_m))) / t_0;
	else
		tmp = ((x - y_m) / hypot(x, y_m)) * (1.0 + (x / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], 2.0], N[(N[(N[(N[(x * y$95$m), $MachinePrecision] - N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(x - y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 99.9%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \frac{\left(x - y\right) \cdot \color{blue}{\left(y + x\right)}}{x \cdot x + y \cdot y} \]

      2. distribute-rgt-in 99.9%

         \[\leadsto \frac{\color{blue}{y \cdot \left(x - y\right) + x \cdot \left(x - y\right)}}{x \cdot x + y \cdot y} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{y \cdot \left(x - y\right) + x \cdot \left(x - y\right)}}{x \cdot x + y \cdot y} \]
   5. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto \frac{y \cdot \color{blue}{\left(x + \left(-y\right)\right)} + x \cdot \left(x - y\right)}{x \cdot x + y \cdot y} \]

      2. distribute-lft-in 99.9%

         \[\leadsto \frac{\color{blue}{\left(y \cdot x + y \cdot \left(-y\right)\right)} + x \cdot \left(x - y\right)}{x \cdot x + y \cdot y} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\left(y \cdot x + y \cdot \left(-y\right)\right)} + x \cdot \left(x - y\right)}{x \cdot x + y \cdot y} \]

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}} \]

      2. times-frac 3.1%

         \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}} \]

      3. hypot-define 3.1%

         \[\leadsto \frac{x - y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}} \]

      4. hypot-define 99.9%

         \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}} \]

   5. Taylor expanded in x around 0 17.0%

      \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \color{blue}{\left(1 + \frac{x}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\frac{\left(x \cdot y - y \cdot y\right) + x \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \left(1 + \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := x \cdot x + y\_m \cdot y\_m\\ \mathbf{if}\;\frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{t\_0} \leq 2:\\ \;\;\;\;\frac{\left(x \cdot y\_m - y\_m \cdot y\_m\right) + x \cdot \left(x - y\_m\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y\_m}}{\frac{y\_m}{x}} + -1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (+ (* x x) (* y_m y_m))))
   (if (<= (/ (* (- x y_m) (+ x y_m)) t_0) 2.0)
     (/ (+ (- (* x y_m) (* y_m y_m)) (* x (- x y_m))) t_0)
     (+ (* 2.0 (/ (/ x y_m) (/ y_m x))) -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = (x * x) + (y_m * y_m);
	double tmp;
	if ((((x - y_m) * (x + y_m)) / t_0) <= 2.0) {
		tmp = (((x * y_m) - (y_m * y_m)) + (x * (x - y_m))) / t_0;
	} else {
		tmp = (2.0 * ((x / y_m) / (y_m / x))) + -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) + (y_m * y_m)
    if ((((x - y_m) * (x + y_m)) / t_0) <= 2.0d0) then
        tmp = (((x * y_m) - (y_m * y_m)) + (x * (x - y_m))) / t_0
    else
        tmp = (2.0d0 * ((x / y_m) / (y_m / x))) + (-1.0d0)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = (x * x) + (y_m * y_m);
	double tmp;
	if ((((x - y_m) * (x + y_m)) / t_0) <= 2.0) {
		tmp = (((x * y_m) - (y_m * y_m)) + (x * (x - y_m))) / t_0;
	} else {
		tmp = (2.0 * ((x / y_m) / (y_m / x))) + -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = (x * x) + (y_m * y_m)
	tmp = 0
	if (((x - y_m) * (x + y_m)) / t_0) <= 2.0:
		tmp = (((x * y_m) - (y_m * y_m)) + (x * (x - y_m))) / t_0
	else:
		tmp = (2.0 * ((x / y_m) / (y_m / x))) + -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(x * x) + Float64(y_m * y_m))
	tmp = 0.0
	if (Float64(Float64(Float64(x - y_m) * Float64(x + y_m)) / t_0) <= 2.0)
		tmp = Float64(Float64(Float64(Float64(x * y_m) - Float64(y_m * y_m)) + Float64(x * Float64(x - y_m))) / t_0);
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(x / y_m) / Float64(y_m / x))) + -1.0);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = (x * x) + (y_m * y_m);
	tmp = 0.0;
	if ((((x - y_m) * (x + y_m)) / t_0) <= 2.0)
		tmp = (((x * y_m) - (y_m * y_m)) + (x * (x - y_m))) / t_0;
	else
		tmp = (2.0 * ((x / y_m) / (y_m / x))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], 2.0], N[(N[(N[(N[(x * y$95$m), $MachinePrecision] - N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(2.0 * N[(N[(x / y$95$m), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 99.9%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \frac{\left(x - y\right) \cdot \color{blue}{\left(y + x\right)}}{x \cdot x + y \cdot y} \]

      2. distribute-rgt-in 99.9%

         \[\leadsto \frac{\color{blue}{y \cdot \left(x - y\right) + x \cdot \left(x - y\right)}}{x \cdot x + y \cdot y} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{y \cdot \left(x - y\right) + x \cdot \left(x - y\right)}}{x \cdot x + y \cdot y} \]
   5. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto \frac{y \cdot \color{blue}{\left(x + \left(-y\right)\right)} + x \cdot \left(x - y\right)}{x \cdot x + y \cdot y} \]

      2. distribute-lft-in 99.9%

         \[\leadsto \frac{\color{blue}{\left(y \cdot x + y \cdot \left(-y\right)\right)} + x \cdot \left(x - y\right)}{x \cdot x + y \cdot y} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\left(y \cdot x + y \cdot \left(-y\right)\right)} + x \cdot \left(x - y\right)}{x \cdot x + y \cdot y} \]

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}} \]

      2. times-frac 3.1%

         \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}} \]

      3. hypot-define 3.1%

         \[\leadsto \frac{x - y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}} \]

      4. hypot-define 99.9%

         \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}} \]
   5. Step-by-step derivation

      1. clear-num 99.9%

         \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}} \]

      2. frac-times 99.9%

         \[\leadsto \color{blue}{\frac{\left(x - y\right) \cdot 1}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}}} \]

      3. *-rgt-identity 99.9%

         \[\leadsto \frac{\color{blue}{x - y}}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}}} \]

   7. Taylor expanded in x around 0 55.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 55.0%

         \[\leadsto \color{blue}{2 \cdot \frac{{x}^{2}}{{y}^{2}} + \left(-1\right)} \]

      2. metadata-eval 55.0%

         \[\leadsto 2 \cdot \frac{{x}^{2}}{{y}^{2}} + \color{blue}{-1} \]

      3. unpow2 55.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}} + -1 \]

      4. unpow2 55.0%

         \[\leadsto 2 \cdot \frac{x \cdot x}{\color{blue}{y \cdot y}} + -1 \]

      5. times-frac 80.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} + -1 \]

      6. unpow2 80.6%

         \[\leadsto 2 \cdot \color{blue}{{\left(\frac{x}{y}\right)}^{2}} + -1 \]

   9. Simplified 80.6%

      \[\leadsto \color{blue}{2 \cdot {\left(\frac{x}{y}\right)}^{2} + -1} \]
   10. Step-by-step derivation

       1. unpow2 80.6%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} + -1 \]

       2. clear-num 80.6%

          \[\leadsto 2 \cdot \left(\frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{y}{x}}}\right) + -1 \]

       3. un-div-inv 80.6%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1 \]

   11. Applied egg-rr 80.6%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\frac{\left(x \cdot y - y \cdot y\right) + x \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{\frac{y}{x}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y\_m}}{\frac{y\_m}{x}} + -1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (- x y_m) (+ x y_m)) (+ (* x x) (* y_m y_m)))))
   (if (<= t_0 2.0) t_0 (+ (* 2.0 (/ (/ x y_m) (/ y_m x))) -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (2.0 * ((x / y_m) / (y_m / x))) + -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = (2.0d0 * ((x / y_m) / (y_m / x))) + (-1.0d0)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (2.0 * ((x / y_m) / (y_m / x))) + -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m))
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = (2.0 * ((x / y_m) / (y_m / x))) + -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(Float64(x - y_m) * Float64(x + y_m)) / Float64(Float64(x * x) + Float64(y_m * y_m)))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(x / y_m) / Float64(y_m / x))) + -1.0);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = (2.0 * ((x / y_m) / (y_m / x))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(N[(2.0 * N[(N[(x / y$95$m), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 99.9%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}} \]

      2. times-frac 3.1%

         \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}} \]

      3. hypot-define 3.1%

         \[\leadsto \frac{x - y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}} \]

      4. hypot-define 99.9%

         \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}} \]
   5. Step-by-step derivation

      1. clear-num 99.9%

         \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}} \]

      2. frac-times 99.9%

         \[\leadsto \color{blue}{\frac{\left(x - y\right) \cdot 1}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}}} \]

      3. *-rgt-identity 99.9%

         \[\leadsto \frac{\color{blue}{x - y}}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}}} \]

   7. Taylor expanded in x around 0 55.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 55.0%

         \[\leadsto \color{blue}{2 \cdot \frac{{x}^{2}}{{y}^{2}} + \left(-1\right)} \]

      2. metadata-eval 55.0%

         \[\leadsto 2 \cdot \frac{{x}^{2}}{{y}^{2}} + \color{blue}{-1} \]

      3. unpow2 55.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}} + -1 \]

      4. unpow2 55.0%

         \[\leadsto 2 \cdot \frac{x \cdot x}{\color{blue}{y \cdot y}} + -1 \]

      5. times-frac 80.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} + -1 \]

      6. unpow2 80.6%

         \[\leadsto 2 \cdot \color{blue}{{\left(\frac{x}{y}\right)}^{2}} + -1 \]

   9. Simplified 80.6%

      \[\leadsto \color{blue}{2 \cdot {\left(\frac{x}{y}\right)}^{2} + -1} \]
   10. Step-by-step derivation

       1. unpow2 80.6%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} + -1 \]

       2. clear-num 80.6%

          \[\leadsto 2 \cdot \left(\frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{y}{x}}}\right) + -1 \]

       3. un-div-inv 80.6%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1 \]

   11. Applied egg-rr 80.6%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{\frac{y}{x}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y\_m}{x} \cdot \frac{y\_m}{x}\right)\\ \mathbf{elif}\;y\_m \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{x + y\_m}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.55e-162)
   (+ 1.0 (* -2.0 (* (/ y_m x) (/ y_m x))))
   (if (<= y_m 5.5e-11)
     (* (- x y_m) (/ (+ x y_m) (+ (* x x) (* y_m y_m))))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.55e-162) {
		tmp = 1.0 + (-2.0 * ((y_m / x) * (y_m / x)));
	} else if (y_m <= 5.5e-11) {
		tmp = (x - y_m) * ((x + y_m) / ((x * x) + (y_m * y_m)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.55d-162) then
        tmp = 1.0d0 + ((-2.0d0) * ((y_m / x) * (y_m / x)))
    else if (y_m <= 5.5d-11) then
        tmp = (x - y_m) * ((x + y_m) / ((x * x) + (y_m * y_m)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.55e-162) {
		tmp = 1.0 + (-2.0 * ((y_m / x) * (y_m / x)));
	} else if (y_m <= 5.5e-11) {
		tmp = (x - y_m) * ((x + y_m) / ((x * x) + (y_m * y_m)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.55e-162:
		tmp = 1.0 + (-2.0 * ((y_m / x) * (y_m / x)))
	elif y_m <= 5.5e-11:
		tmp = (x - y_m) * ((x + y_m) / ((x * x) + (y_m * y_m)))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.55e-162)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y_m / x) * Float64(y_m / x))));
	elseif (y_m <= 5.5e-11)
		tmp = Float64(Float64(x - y_m) * Float64(Float64(x + y_m) / Float64(Float64(x * x) + Float64(y_m * y_m))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.55e-162)
		tmp = 1.0 + (-2.0 * ((y_m / x) * (y_m / x)));
	elseif (y_m <= 5.5e-11)
		tmp = (x - y_m) * ((x + y_m) / ((x * x) + (y_m * y_m)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.55e-162], N[(1.0 + N[(-2.0 * N[(N[(y$95$m / x), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 5.5e-11], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(x + y$95$m), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 1.5499999999999999e-162

   1. Initial program 71.1%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. associate-/l* 71.6%

         \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 27.3%

      \[\leadsto \color{blue}{1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 27.3%

         \[\leadsto 1 + -2 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 27.3%

         \[\leadsto 1 + -2 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 33.4%

         \[\leadsto 1 + -2 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   7. Applied egg-rr 33.4%

      \[\leadsto 1 + -2 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   if 1.5499999999999999e-162 < y < 5.49999999999999975e-11

   1. Initial program 99.9%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. associate-/l* 96.6%

         \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]

   3. Simplified 96.6%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
   4. Add Preprocessing

   if 5.49999999999999975e-11 < y

   1. Initial program 100.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.0%

         \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.8 \cdot 10^{-165} \lor \neg \left(y\_m \leq 1.45 \cdot 10^{-111}\right) \land y\_m \leq 4.3 \cdot 10^{-97}:\\ \;\;\;\;\frac{x - y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (or (<= y_m 1.8e-165) (and (not (<= y_m 1.45e-111)) (<= y_m 4.3e-97)))
   (/ (- x y_m) x)
   -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m <= 1.8e-165) || (!(y_m <= 1.45e-111) && (y_m <= 4.3e-97))) {
		tmp = (x - y_m) / x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((y_m <= 1.8d-165) .or. (.not. (y_m <= 1.45d-111)) .and. (y_m <= 4.3d-97)) then
        tmp = (x - y_m) / x
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if ((y_m <= 1.8e-165) || (!(y_m <= 1.45e-111) && (y_m <= 4.3e-97))) {
		tmp = (x - y_m) / x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if (y_m <= 1.8e-165) or (not (y_m <= 1.45e-111) and (y_m <= 4.3e-97)):
		tmp = (x - y_m) / x
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if ((y_m <= 1.8e-165) || (!(y_m <= 1.45e-111) && (y_m <= 4.3e-97)))
		tmp = Float64(Float64(x - y_m) / x);
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((y_m <= 1.8e-165) || (~((y_m <= 1.45e-111)) && (y_m <= 4.3e-97)))
		tmp = (x - y_m) / x;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[Or[LessEqual[y$95$m, 1.8e-165], And[N[Not[LessEqual[y$95$m, 1.45e-111]], $MachinePrecision], LessEqual[y$95$m, 4.3e-97]]], N[(N[(x - y$95$m), $MachinePrecision] / x), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.79999999999999992e-165 or 1.45000000000000001e-111 < y < 4.3e-97

   1. Initial program 71.9%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. associate-/l* 72.3%

         \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]

   3. Simplified 72.3%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 31.6%

      \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1}{x}} \]
   6. Step-by-step derivation

      1. un-div-inv 31.7%

         \[\leadsto \color{blue}{\frac{x - y}{x}} \]

   7. Applied egg-rr 31.7%

      \[\leadsto \color{blue}{\frac{x - y}{x}} \]

   if 1.79999999999999992e-165 < y < 1.45000000000000001e-111 or 4.3e-97 < y

   1. Initial program 97.7%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. associate-/l* 94.5%

         \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]

   3. Simplified 94.5%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 73.2%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-165} \lor \neg \left(y \leq 1.45 \cdot 10^{-111}\right) \land y \leq 4.3 \cdot 10^{-97}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 5.6 \cdot 10^{-165}:\\ \;\;\;\;\frac{x - y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y\_m}\right) \cdot \left(\frac{x}{y\_m} + -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 5.6e-165)
   (/ (- x y_m) x)
   (* (+ 1.0 (/ x y_m)) (+ (/ x y_m) -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 5.6e-165) {
		tmp = (x - y_m) / x;
	} else {
		tmp = (1.0 + (x / y_m)) * ((x / y_m) + -1.0);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 5.6d-165) then
        tmp = (x - y_m) / x
    else
        tmp = (1.0d0 + (x / y_m)) * ((x / y_m) + (-1.0d0))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 5.6e-165) {
		tmp = (x - y_m) / x;
	} else {
		tmp = (1.0 + (x / y_m)) * ((x / y_m) + -1.0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 5.6e-165:
		tmp = (x - y_m) / x
	else:
		tmp = (1.0 + (x / y_m)) * ((x / y_m) + -1.0)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 5.6e-165)
		tmp = Float64(Float64(x - y_m) / x);
	else
		tmp = Float64(Float64(1.0 + Float64(x / y_m)) * Float64(Float64(x / y_m) + -1.0));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 5.6e-165)
		tmp = (x - y_m) / x;
	else
		tmp = (1.0 + (x / y_m)) * ((x / y_m) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 5.6e-165], N[(N[(x - y$95$m), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5999999999999999e-165

   1. Initial program 71.5%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. associate-/l* 71.9%

         \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 31.1%

      \[\leadsto \left(x - y\right) \cdot \color{blue}{\frac{1}{x}} \]
   6. Step-by-step derivation

      1. un-div-inv 31.2%

         \[\leadsto \color{blue}{\frac{x - y}{x}} \]

   7. Applied egg-rr 31.2%

      \[\leadsto \color{blue}{\frac{x - y}{x}} \]

   if 5.5999999999999999e-165 < y

   1. Initial program 97.9%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 97.8%

         \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}} \]

      2. times-frac 95.0%

         \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}} \]

      3. hypot-define 95.2%

         \[\leadsto \frac{x - y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}} \]

      4. hypot-define 99.8%

         \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}} \]

   5. Taylor expanded in x around 0 72.7%

      \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \color{blue}{\left(1 + \frac{x}{y}\right)} \]

   6. Taylor expanded in x around 0 72.2%

      \[\leadsto \color{blue}{\left(\frac{x}{y} - 1\right)} \cdot \left(1 + \frac{x}{y}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-165}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 6.7 \cdot 10^{-164}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y\_m}{x} \cdot \frac{y\_m}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y\_m}\right) \cdot \left(\frac{x}{y\_m} + -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 6.7e-164)
   (+ 1.0 (* -2.0 (* (/ y_m x) (/ y_m x))))
   (* (+ 1.0 (/ x y_m)) (+ (/ x y_m) -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 6.7e-164) {
		tmp = 1.0 + (-2.0 * ((y_m / x) * (y_m / x)));
	} else {
		tmp = (1.0 + (x / y_m)) * ((x / y_m) + -1.0);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 6.7d-164) then
        tmp = 1.0d0 + ((-2.0d0) * ((y_m / x) * (y_m / x)))
    else
        tmp = (1.0d0 + (x / y_m)) * ((x / y_m) + (-1.0d0))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 6.7e-164) {
		tmp = 1.0 + (-2.0 * ((y_m / x) * (y_m / x)));
	} else {
		tmp = (1.0 + (x / y_m)) * ((x / y_m) + -1.0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 6.7e-164:
		tmp = 1.0 + (-2.0 * ((y_m / x) * (y_m / x)))
	else:
		tmp = (1.0 + (x / y_m)) * ((x / y_m) + -1.0)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 6.7e-164)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y_m / x) * Float64(y_m / x))));
	else
		tmp = Float64(Float64(1.0 + Float64(x / y_m)) * Float64(Float64(x / y_m) + -1.0));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 6.7e-164)
		tmp = 1.0 + (-2.0 * ((y_m / x) * (y_m / x)));
	else
		tmp = (1.0 + (x / y_m)) * ((x / y_m) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 6.7e-164], N[(1.0 + N[(-2.0 * N[(N[(y$95$m / x), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.69999999999999999e-164

   1. Initial program 71.5%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. associate-/l* 71.9%

         \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 27.5%

      \[\leadsto \color{blue}{1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 27.5%

         \[\leadsto 1 + -2 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 27.5%

         \[\leadsto 1 + -2 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 33.5%

         \[\leadsto 1 + -2 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   7. Applied egg-rr 33.5%

      \[\leadsto 1 + -2 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   if 6.69999999999999999e-164 < y

   1. Initial program 97.9%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 97.8%

         \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}} \]

      2. times-frac 95.0%

         \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}} \]

      3. hypot-define 95.2%

         \[\leadsto \frac{x - y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}} \]

      4. hypot-define 99.8%

         \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}} \]

   5. Taylor expanded in x around 0 72.7%

      \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \color{blue}{\left(1 + \frac{x}{y}\right)} \]

   6. Taylor expanded in x around 0 72.2%

      \[\leadsto \color{blue}{\left(\frac{x}{y} - 1\right)} \cdot \left(1 + \frac{x}{y}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.7 \cdot 10^{-164}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) \cdot \left(\frac{x}{y} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 3.4 \cdot 10^{-164}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y\_m}{x} \cdot \frac{y\_m}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y\_m}}{\frac{y\_m}{x}} + -1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 3.4e-164)
   (+ 1.0 (* -2.0 (* (/ y_m x) (/ y_m x))))
   (+ (* 2.0 (/ (/ x y_m) (/ y_m x))) -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 3.4e-164) {
		tmp = 1.0 + (-2.0 * ((y_m / x) * (y_m / x)));
	} else {
		tmp = (2.0 * ((x / y_m) / (y_m / x))) + -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 3.4d-164) then
        tmp = 1.0d0 + ((-2.0d0) * ((y_m / x) * (y_m / x)))
    else
        tmp = (2.0d0 * ((x / y_m) / (y_m / x))) + (-1.0d0)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 3.4e-164) {
		tmp = 1.0 + (-2.0 * ((y_m / x) * (y_m / x)));
	} else {
		tmp = (2.0 * ((x / y_m) / (y_m / x))) + -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 3.4e-164:
		tmp = 1.0 + (-2.0 * ((y_m / x) * (y_m / x)))
	else:
		tmp = (2.0 * ((x / y_m) / (y_m / x))) + -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 3.4e-164)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y_m / x) * Float64(y_m / x))));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(x / y_m) / Float64(y_m / x))) + -1.0);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 3.4e-164)
		tmp = 1.0 + (-2.0 * ((y_m / x) * (y_m / x)));
	else
		tmp = (2.0 * ((x / y_m) / (y_m / x))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 3.4e-164], N[(1.0 + N[(-2.0 * N[(N[(y$95$m / x), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(x / y$95$m), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.4e-164

   1. Initial program 71.5%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. associate-/l* 71.9%

         \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 27.5%

      \[\leadsto \color{blue}{1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 27.5%

         \[\leadsto 1 + -2 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 27.5%

         \[\leadsto 1 + -2 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 33.5%

         \[\leadsto 1 + -2 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   7. Applied egg-rr 33.5%

      \[\leadsto 1 + -2 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   if 3.4e-164 < y

   1. Initial program 97.9%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 97.8%

         \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}} \]

      2. times-frac 95.0%

         \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}} \]

      3. hypot-define 95.2%

         \[\leadsto \frac{x - y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}} \]

      4. hypot-define 99.8%

         \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\color{blue}{\mathsf{hypot}\left(x, y\right)}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}} \]
   5. Step-by-step derivation

      1. clear-num 99.8%

         \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}} \]

      2. frac-times 99.9%

         \[\leadsto \color{blue}{\frac{\left(x - y\right) \cdot 1}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}}} \]

      3. *-rgt-identity 99.9%

         \[\leadsto \frac{\color{blue}{x - y}}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}}} \]

   7. Taylor expanded in x around 0 70.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 70.9%

         \[\leadsto \color{blue}{2 \cdot \frac{{x}^{2}}{{y}^{2}} + \left(-1\right)} \]

      2. metadata-eval 70.9%

         \[\leadsto 2 \cdot \frac{{x}^{2}}{{y}^{2}} + \color{blue}{-1} \]

      3. unpow2 70.9%

         \[\leadsto 2 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}} + -1 \]

      4. unpow2 70.9%

         \[\leadsto 2 \cdot \frac{x \cdot x}{\color{blue}{y \cdot y}} + -1 \]

      5. times-frac 72.9%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} + -1 \]

      6. unpow2 72.9%

         \[\leadsto 2 \cdot \color{blue}{{\left(\frac{x}{y}\right)}^{2}} + -1 \]

   9. Simplified 72.9%

      \[\leadsto \color{blue}{2 \cdot {\left(\frac{x}{y}\right)}^{2} + -1} \]
   10. Step-by-step derivation

       1. unpow2 72.9%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} + -1 \]

       2. clear-num 72.9%

          \[\leadsto 2 \cdot \left(\frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{y}{x}}}\right) + -1 \]

       3. un-div-inv 72.9%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1 \]

   11. Applied egg-rr 72.9%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-164}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{\frac{y}{x}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{-165}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= y_m 1e-165) 1.0 -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1e-165) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1d-165) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1e-165) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1e-165:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1e-165)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1e-165)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1e-165], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1e-165

   1. Initial program 71.5%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. associate-/l* 71.9%

         \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 31.6%

      \[\leadsto \color{blue}{1} \]

   if 1e-165 < y

   1. Initial program 97.9%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. associate-/l* 94.8%

         \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]

   3. Simplified 94.8%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 70.8%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-165}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.1% accurate, 15.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ -1 \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 -1.0)

C

y_m = fabs(y);
double code(double x, double y_m) {
	return -1.0;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = -1.0d0
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return -1.0;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return -1.0

Julia

y_m = abs(y)
function code(x, y_m)
	return -1.0
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = -1.0;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := -1.0

Derivation

1. Initial program 76.5%

   \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
2. Step-by-step derivation

   1. associate-/l* 76.3%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]

3. Simplified 76.3%

   \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 68.9%

   \[\leadsto \color{blue}{-1} \]

6. Final simplification 68.9%

   \[\leadsto -1 \]
7. Add Preprocessing

Developer target: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;0.5 < t\_0 \land t\_0 < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (and (< 0.5 t_0) (< t_0 2.0))
     (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))
     (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = fabs((x / y));
	double tmp;
	if ((0.5 < t_0) && (t_0 < 2.0)) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y))
    if ((0.5d0 < t_0) .and. (t_0 < 2.0d0)) then
        tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
    else
        tmp = 1.0d0 - (2.0d0 / (1.0d0 + ((x / y) * (x / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.abs((x / y));
	double tmp;
	if ((0.5 < t_0) && (t_0 < 2.0)) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.fabs((x / y))
	tmp = 0
	if (0.5 < t_0) and (t_0 < 2.0):
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
	else:
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))))
	return tmp

Julia

function code(x, y)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if ((0.5 < t_0) && (t_0 < 2.0))
		tmp = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = Float64(1.0 - Float64(2.0 / Float64(1.0 + Float64(Float64(x / y) * Float64(x / y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs((x / y));
	tmp = 0.0;
	if ((0.5 < t_0) && (t_0 < 2.0))
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	else
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[And[Less[0.5, t$95$0], Less[t$95$0, 2.0]], N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(2.0 / N[(1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024048 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :alt
  (if (and (< 0.5 (fabs (/ x y))) (< (fabs (/ x y)) 2.0)) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))